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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1omo | Structured version Visualization version GIF version |
Description: There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 47526 assuming ax-un 7725 (see f1omoALT 47528). (Contributed by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
f1omo.1 | ⊢ (𝜑 → 𝐹 = (𝐴 × {1o})) |
Ref | Expression |
---|---|
f1omo | ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 8476 | . . . 4 ⊢ 1o ∈ V | |
2 | eqid 2733 | . . . 4 ⊢ ((𝐴 × {1o})‘𝑋) = ((𝐴 × {1o})‘𝑋) | |
3 | 1, 2 | fvconst0ci 47525 | . . 3 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) |
4 | mo0 47498 | . . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)) | |
5 | el1o 8495 | . . . . . . . 8 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅) | |
6 | el1o 8495 | . . . . . . . 8 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅) | |
7 | eqtr3 2759 | . . . . . . . 8 ⊢ ((𝑦 = ∅ ∧ 𝑥 = ∅) → 𝑦 = 𝑥) | |
8 | 5, 6, 7 | syl2anb 599 | . . . . . . 7 ⊢ ((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥) |
9 | 8 | gen2 1799 | . . . . . 6 ⊢ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥) |
10 | eleq1w 2817 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 1o ↔ 𝑥 ∈ 1o)) | |
11 | 10 | mo4 2561 | . . . . . 6 ⊢ (∃*𝑦 𝑦 ∈ 1o ↔ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥)) |
12 | 9, 11 | mpbir 230 | . . . . 5 ⊢ ∃*𝑦 𝑦 ∈ 1o |
13 | eleq2w2 2729 | . . . . . 6 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ 𝑦 ∈ 1o)) | |
14 | 13 | mobidv 2544 | . . . . 5 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ ∃*𝑦 𝑦 ∈ 1o)) |
15 | 12, 14 | mpbiri 258 | . . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)) |
16 | 4, 15 | jaoi 856 | . . 3 ⊢ ((((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)) |
17 | 3, 16 | ax-mp 5 | . 2 ⊢ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋) |
18 | f1omo.1 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝐴 × {1o})) | |
19 | 18 | fveq1d 6894 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋)) |
20 | 19 | eleq2d 2820 | . . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝑋) ↔ 𝑦 ∈ ((𝐴 × {1o})‘𝑋))) |
21 | 20 | mobidv 2544 | . 2 ⊢ (𝜑 → (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))) |
22 | 17, 21 | mpbiri 258 | 1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∨ wo 846 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ∃*wmo 2533 ∅c0 4323 {csn 4629 × cxp 5675 ‘cfv 6544 1oc1o 8459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-1o 8466 |
This theorem is referenced by: indthinc 47672 prsthinc 47674 |
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